Finite Group Databasebeta

← #14542401

Subgroups — #14542401

Order 2176.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763535C211yesmaximal, characteristic, complemented
2#11763531C213yesmaximal, complemented
3#11763500C213yesmaximal, complemented
4#11763494C213yesmaximal, complemented
5Q8×C22⋊D17C211yesmaximal, characteristic, complemented
6#11763471C211yesmaximal, characteristic, complemented
7#11763470C211yesmaximal, characteristic, complemented
8#11763460C213yesmaximal, complemented
9#11763363C213yesmaximal, complemented
10#11763346C213yesmaximal, complemented
11#11763343C213yesmaximal, complemented
12C4×#63256C211yesmaximal, characteristic, complemented
13C4×#63252C213yesmaximal, complemented
14C4×Q8×D17C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C17C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C2×#63252C2213yes—
20C4×Q8×C17C2211yescharacteristic
21C4×Q8×C17C2213yes—
22C4×Q8×C17C2211yescharacteristic
23C4×Q8×C17C2213yes—
24C2×C4⋊C4×C17C2213yes—
25C2×C42×C17C2213yes—
26#10587633C2213yes—
27#10587632C2213yes—
28#10587632C2213yes—
29Q8×Dic17C2211yescharacteristic
30Q8×Dic17C2211yescharacteristic
31#10587630C2213yes—
32#10587616C2213yes—
33#10587615C2216yes—
34C4×C22⋊D17C2213yes—
35C4×C22⋊D17C2213yes—
36C4×C22⋊D17C2211yescharacteristic
37C4×C22⋊D17C2213yes—
38#10587612C2213yes—
39#10587610C2213yes—
40#10587584C2216yes—
41#10587583C2213yes—
42#10587582C2216yes—
43#10587581C2213yes—
44#10587580C2216yes—
45#10587579C2213yes—
46#10587578C2213yes—
47C4⋊C4×D17C2213yes—
48#10587576C2213yes—
49#10587575C2216yes—
50#10587574C2213yes—
51#10587573C2213yes—
52#10587562C2216yes—
53#10587561C2213yes—
54C4×D68C2213yes—
55C4×D68C2213yes—
56C4×D68C2213yes—
57#10587558C2213yes—
58#10587558C2213yes—
59#10587558C2216yes—
60C42×D17C2213yes—
61C42×D17C2213yes—
62#10587556C2213yes—
63C4×Dic34C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66C2×Q8×C17—24no—
67C2×Q8×C17C2313yes—
68C2×Q8×C17C2311yescharacteristic
69C22×C4×C17C2313yes—
70C22×C4×C17C2313yes—
71C22×C4×C17C2311yescharacteristic
72C2×C22⋊D17C2313yes—
73C2×C22⋊D17C2311yescharacteristic
74#63256—24no—
75Q8×D17—24no—
76#63252—212no—
77C2×D68C2313yes—
78C2×D68C2313yes—
79C2×C4×D17C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2313yes—
82C2×C4×D17C2313yes—
83C2×C4×D17C2311yescharacteristic
84C2×C4×D17C2313yes—
85C2×Dic34C2313yes—
86C2×Dic34C2313yes—
87C4⋊C4×C17C2313yes—
88C4⋊C4×C17C2313yes—
89C4⋊C4×C17C2316yes—
90C42×C17C2313yes—
91C42×C17C2313yes—
92C42×C17C2313yes—
93C42×C17C2313yes—
94C22⋊Dic17C2313yes—
95C22⋊Dic17C2311yescharacteristic
96#63227C2311yescharacteristic
97#63227C2313yes—
98#63227C2313yes—
99#63227C2313yes—
100#63227C2316yes—
101#63227C2316yes—
102#63227C2316yes—
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2313yes—
107#63225C2311yescharacteristic
108#63225C2313yes—
109#63225C2313yes—
110#63225C2313yes—
111#63225C2316yes—
112#63225C2316yes—
113#63225C2316yes—
114C4×Dic17C2311yescharacteristic
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C4×Dic17C2313yes—
120C23×C17C2411yescharacteristic
121C22×D17C2413yes—
122C22×D17C2411yescharacteristic
123Q8×C17—26no—
124Q8×C17#4114yescomplemented
125C2×C4×C17—26no—
126C2×C4×C17—26no—
127C2×C4×C17—22no—
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2413yes—
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—26no—
135C22⋊D17#4114yescomplemented
136C2×Dic17C2413yes—
137C2×Dic17C2411yescharacteristic
138C2×Dic17C2411yescharacteristic
139C2×Dic17C2413yes—
140C2×Dic17C2413yes—
141C2×Dic17C2411yescharacteristic
142D68—26no—
143D68—26no—
144C4×D17—26no—
145C4×D17—26no—
146C4×D17—26no—
147C4×D17—22no—
148C4×D17—26no—
149C4×D17—26no—
150Dic34—26no—
151Dic34—26no—
152#3482—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17C2×#4112yes—
156C22×C17C2511yescharacteristic
157C2×D17—21no—
158C2×D17—21no—
159C2×D17—23no—
160C2×D17—23no—
161C2×D17—23no—
162C2×D17C2×#4112yes—
163C4×C17—23no—
164C4×C17—23no—
165C4×C17—23no—
166C4×C17—21no—
167C4×C17C2×#4116yes—
168C4×C172−1+412yes—
169Dic17—23no—
170Dic17—23no—
171Dic17—21no—
172Dic17—21no—
173Dic17—23no—
174Dic17C2×#4112yes—
175#557—171no—
176#556—173no—
177#552—173no—
178#552—173no—
179D4×Q8—171no—
180#548—171no—
181#548—171no—
182#547—173no—
183#533—173no—
184#529—173no—
185#529—173no—
186C4×#41—171no—
187C4×#41—173no—
188C2×C4×Q8—171no—
189C2×C4×Q8D1711yescharacteristic, complemented
190C2×C17—22no—
191C2×C17C22×#4111yescharacteristic
192C2×C17C22×#4111yescharacteristic
193C2×C17C2×2−1+411yescharacteristic
194D17—22no—
195D17—43no—
196C2×#41—171no—
197C2×#41—173no—
198C22×Q8—171no—
199C22×Q8C2×D1711yescharacteristic
200C4⋊Q8—173no—
201C4⋊Q8—173no—
202#126—176no—
203#126—176no—
204#125—173no—
205#125—176no—
206#125—173no—
207#124—173no—
208#124—173no—
209#123—176no—
210#123—176no—
211C22⋊Q8—173no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—176no—
215C22⋊Q8—173no—
216#121—173no—
217#121—173no—
218C4×Q8—171no—
219C4×Q8—171no—
220C4×Q8—173no—
221C4×Q8—173no—
222C4×Q8—173no—
223C4×Q8—173no—
224C4×Q8C2×D1711yescharacteristic
225C4×Q8C2×D1713yes—
226C4×Q8C2×D1711yescharacteristic
227C4×Q8C2×D1713yes—
228C4×D4—171no—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—173no—
236#117—173no—
237#117—173no—
238#117—173no—
239#117—173no—
240#117—176no—
241C2×C4⋊C4—173no—
242C2×C4⋊C4C2×D1713yes—
243C2×C42—173no—
244C2×C42—173no—
245C2×C42C2×D1713yes—
246C17#348211yescharacteristic, complemented, Sylow
247#41—3412no—
248#41—344no—
249C2×Q8—24no—
250C2×Q8—173no—
251C2×Q8—173no—
252C2×Q8—344no—
253C2×Q8C22×D1713yes—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—171no—
256C2×D4—173no—
257C2×D4—173no—
258C2×D4—173no—
259C22×C4—173no—
260C22×C4—173no—
261C22×C4—173no—
262C22×C4—173no—
263C22×C4—171no—
264C22×C4—173no—
265C22×C4C22×D1713yes—
266C22×C4C22×D1713yes—
267C22×C4C22×D1711yescharacteristic
268C4⋊C4—173no—
269C4⋊C4—176no—
270C4⋊C4—171no—
271C4⋊C4—173no—
272C4⋊C4—173no—
273C4⋊C4—173no—
274C4⋊C4—173no—
275C4⋊C4—173no—
276C4⋊C4—173no—
277C4⋊C4—176no—
278C4⋊C4—176no—
279C4⋊C4C22×D1713yes—
280C4⋊C4C22×D1713yes—
281C4⋊C4C22×D1716yes—
282C22⋊C4—171no—
283C22⋊C4—173no—
284C22⋊C4—171no—
285C22⋊C4—173no—
286C22⋊C4—176no—
287C22⋊C4—173no—
288C22⋊C4—173no—
289C22⋊C4—176no—
290C22⋊C4—176no—
291C42—173no—
292C42—173no—
293C42—173no—
294C42—173no—
295C42—171no—
296C42—173no—
297C42C22×D1713yes—
298C42C22×D1713yes—
299C42C22×D1713yes—
300C42C22×D1713yes—
301C23—171no—
302C23—173no—
303C23C23×D1711yescharacteristic
304Q8—26no—
305Q8—346no—
306Q8—346no—
307Q8#6325214yescomplemented
308D4—174no—
309D4—346no—
310D4—346no—
311D4—346no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—171no—
316C2×C4—171no—
317C2×C4—171no—
318C2×C4—173no—
319C2×C4—173no—
320C2×C4—173no—
321C2×C4—346no—
322C2×C4—346no—
323C2×C4—346no—
324C2×C4—346no—
325C2×C4—342no—
326C2×C4—346no—
327C2×C4C23×D1713yes—
328C2×C4C23×D1711yescharacteristic
329C2×C4C23×D1713yes—
330C2×C4C23×D1713yes—
331C2×C4C23×D1713yes—
332C2×C4C23×D1711yescharacteristic
333C22—21no—
334C22—21no—
335C22—172no—
336C22—341no—
337C22—341no—
338C22—343no—
339C22—343no—
340C22—343no—
341C22#41×D1712yes—
342C22C24×D1711yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—172no—
348C4—343no—
349C4—341no—
350C4—343no—
351C4—341no—
352C4—343no—
353C4C2×#6325216yes—
354C4#1058769412yes—
355C2—22no—
356C2—342no—
357C2—683no—
358C2C2×#41×D1711yescharacteristic
359C2C22×#6325211yescharacteristic
360C2C2×#1058769411yescharacteristic
subgroupG/H
centerC2×C4C23×D17
commutator, socle, cosocleC22×C17C25
FrattiniC22C24×D17
FittingC2×C4×Q8×C17C2
hypercenterC2×C4×Q8D17
hypocenterC17#3482
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11763535C2semidirect
2. #11763531C2semidirect
3. #11763500C2semidirect
4. #11763494C2semidirect
5. Q8×C22⋊D17C2semidirect
6. #11763471C2semidirect
7. #11763470C2semidirect
8. #11763460C2semidirect
9. #11763363C2semidirect
10. #11763346C2semidirect
11. #11763343C2semidirect
12. C4×#63256C2semidirect
13. C4×#63252C2semidirect
14. C4×Q8×D17C2semidirect
15. C2×C4×Q8×C17C2semidirect
124. Q8×C17#41semidirect
135. C22⋊D17#41semidirect
189. C2×C4×Q8D17semidirect
246. C17#3482semidirect
307. Q8#63252semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117635352
#117635312
#117635002
#117634942
Q8×C22⋊D172
#117634712
#117634702
#117634602
#117633632
#117633462
#117633432
C4×#632562
C4×#632522
C4×Q8×D172
C2×C4×Q8×C172
#348217